Kostka polynomials and affine Kac-Moody algebras

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The PI proposes to study the representation theory and combinatoricsunderlying certain generalizations of Hall-Littlewood and Kostka-Foulkes polynomials for affine (and more generally symmetrizable) Kac-Moody algebras. Specifically, the PI wishes to study affine Kostka-Foulkes polynomials from the perspectives of Cherednik-Macdonald theory, q-hypergeometric series and positivity. The PI also hopes to use the polynomials corresponding to arbitrary Kac-Moody algebras toshed more light on representations of these poorly understood Lie algebras.Symmetric functions are multivariate polynomials which remain unchangedwhen the variables are permuted. They admit a rich theory involving an interplay of algebra, combinatorics and representation theory. Hall-Littlewood polynomials are symmetric functions that depend on an extra parameter and interpolate between two very important classes of symmetric functions. The theory of Hall-Littlewood polynomials traces its origins to works of Hall, Littlewood and later of Macdonald. These polynomials occur naturally in diverse areas of mathematics includingalgebra, geometry, representation theory, combinatorics and mathematical physics. They have many important properties and applications.The goal of the proposed research is to study the generalization of the classical Hall-Littlewood polynomials to the case of infinite dimensional (Kac-Moody) Lie algebras and to derive analogs of classical properties in this general setting. The PI hopes to clarify the relation between the generalizations of Hall-Littlewood polynomials on one hand and the generalizations of the classical objects it is related to (such as the "double affine Hecke algebra").